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Supergravity @ 40, GGI Florence, October 27th, 2016
Non-Associative Flux Algebra in String and M-theory from Octonions
DIETER LÜST (LMU, MPI)
Donnerstag, 27. Oktober 16
1
Supergravity @ 40, GGI Florence, October 27th, 2016
In collaboration with M. Günaydin & E. Malek, arXiv:1607.06474
Non-Associative Flux Algebra in String and M-theory from Octonions
DIETER LÜST (LMU, MPI)
Donnerstag, 27. Oktober 16
2
This talk is dedicated to my friend Ioannis Bakas
Donnerstag, 27. Oktober 16
Outline:
3
II) Non-associative R-flux algebra for closed strings
I) Introduction
III) R-flux algebra from octonions
IV) M-theory up-lift of R-flux background
V) Non-associative R-flux algebra in M-theory
Donnerstag, 27. Oktober 16
Point particles in classical Einstein gravity „see“ continuous Riemannian manifolds.
Geometry in general depends on, with what kind of objects you test it.
Strings may see space-time in a different way.
4
We expect the emergence of a new kind of stringy geometry.
I) Introduction
- [xi, x
j ] = 0
Donnerstag, 27. Oktober 16
5
Closed strings in non-geometric R-flux backgrounds⇒ non-associative phase space algebra:
R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.
D.L., arXiv:1010.1361;⇥x
i, x
j⇤
= i
l
3s
~ R
ijkpk
⇥x
i, p
j⇤
= i~�
ij,
⇥p
i, p
j⇤
= 0
=)⇥x
i, x
j, x
k⇤⌘ 1
3⇥⇥
x
1, x
2⇤, x
3⇤+ cycl. perm. = l
3sR
ijk
Donnerstag, 27. Oktober 16
5
Closed strings in non-geometric R-flux backgrounds⇒ non-associative phase space algebra:
R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.
D.L., arXiv:1010.1361;⇥x
i, x
j⇤
= i
l
3s
~ R
ijkpk
⇥x
i, p
j⇤
= i~�
ij,
⇥p
i, p
j⇤
= 0
=)⇥x
i, x
j, x
k⇤⌘ 1
3⇥⇥
x
1, x
2⇤, x
3⇤+ cycl. perm. = l
3sR
ijk
This algebra can be derived from closed string CFT.R. Blumenhagen, A. Deser, D.L. , E. Plauschinn, F. Rennecke, arXiv:1106.0316
C. Condeescu, I. Florakis, D. L., arXiv:1202.6366
C. Blair, arXiv:1405.2283
D. Andriot, M. Larfors, D.L. , P. Patalong:arXiv:1211.6437
I. Bakas, D.L., arXiv:1505.04004
Donnerstag, 27. Oktober 16
5
Closed strings in non-geometric R-flux backgrounds⇒ non-associative phase space algebra:
R. Blumenhagen, E. Plauschinn, arXiv:1010.1263.
D.L., arXiv:1010.1361;⇥x
i, x
j⇤
= i
l
3s
~ R
ijkpk
⇥x
i, p
j⇤
= i~�
ij,
⇥p
i, p
j⇤
= 0
=)⇥x
i, x
j, x
k⇤⌘ 1
3⇥⇥
x
1, x
2⇤, x
3⇤+ cycl. perm. = l
3sR
ijk
This algebra can be derived from closed string CFT.R. Blumenhagen, A. Deser, D.L. , E. Plauschinn, F. Rennecke, arXiv:1106.0316
C. Condeescu, I. Florakis, D. L., arXiv:1202.6366
C. Blair, arXiv:1405.2283
D. Andriot, M. Larfors, D.L. , P. Patalong:arXiv:1211.6437
I. Bakas, D.L., arXiv:1505.04004
This algebra is also closely related to double field theory.R. Blumenhagen, M. Fuchs, F. Hassler, D.L. , R. Sun, arXiv:1312.0719
Donnerstag, 27. Oktober 16
6
Two questions:
Donnerstag, 27. Oktober 16
6
Two questions:
On the mathematical side:
How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?
Donnerstag, 27. Oktober 16
6
Two questions:
On the mathematical side:
How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?
Can one lift the R-flux algebra of closed strings to M-theory?
On the physics side:
Donnerstag, 27. Oktober 16
6
Two questions:
On the mathematical side:
How is the R-flux algebra related to other known non-associative algebras, in particular to the algebra of the octonions?
Can one lift the R-flux algebra of closed strings to M-theory?
On the physics side:
Our conjecture:the answers to these two questions are closely related
Donnerstag, 27. Oktober 16
7
II) Non-geometric string flux backgrounds
Three-dimensional string flux backgrounds:
HijkTi�! f i
jkTj�! Qij
kTk�! Rijk , (i, j, k = 1, . . . , 3)
Chain of three T-duality transformations:
ds
2 = (dx
1)2 + (dx
2)2 + (dx
3)2 , B12 = Nx
3
H123 = NH-flux:
(Hellerman, McGreevy, Williams (2002); C. Hull (2004); Shelton, Taylor, Wecht (2005); Dabholkar, Hull, 2005)
(i) with H-flux:T 3
Donnerstag, 27. Oktober 16
8
ds
2 =�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 , B2 = 0
Globally defined 1-forms:
⌘
1 = dx
1 �Nx
3dx
2, ⌘
2 = dx
2, ⌘
3 = dx
3
d⌘i = f ijk⌘j ^ ⌘k
Geometric flux: f123 = N
x
1(ii) Twisted torus tilde : T-duality along T 3
is a U(1) bundle over :T 3 T 2
Donnerstag, 27. Oktober 16
9
(iii) Q-flux background:
ds
2 =(dx
1)2 + (dx
2)2
1 + N
2 (x3)2+ (dx
3)2 , B23 =Nx
3
1 + N
2 (x3)2
This background is globally not well defined, but it is patched together by a T-duality transformation.
C. Hull (2004)
T-duality along x
2
⇒ T - fold
Donnerstag, 27. Oktober 16
10
(iii) Q-flux background:
ds
2 =(dx
1)2 + (dx
2)2
1 + N
2 (x3)2+ (dx
3)2 , B23 =Nx
3
1 + N
2 (x3)2
This background is globally not well defined, but it is patched together by T-duality transformation.
To make it well defined use double field theory:
Coordinates: (x1, x
2, x
3; x1, x2, x3)
C. Hull (2004)
W. Siegel (1993); C. Hull, B. Zwiebach (2009); C. Hull, O. Hohm, B. Zwiebach (2010,...)
SO(3,3) double field theory:
T-duality along x
2
⇒ T - fold
Donnerstag, 27. Oktober 16
11
M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012);R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013);D. Andriot, A. Betz (2013)
The dual background can then by described by „dual“ metric and a bi-vector:
Bij(x) ! �
ij(x) =12
⇣(g �B)�1 � (g + B)�1
⌘,
g(x) ! g(x) =12
⇣(g �B)�1 + (g + B)�1
⌘�1.
T ij
T ij
Qijk = @k�ij
Donnerstag, 27. Oktober 16
11
M. Grana, R. Minasian, M. Petrini, D. Waldram (2008); D. Andriot, O. Hohm, M. Larfors, D.L., P. Patalong (2011,2012);R. Blumenhagen, A. Deser, E. Plauschinn, F. Rennecke, C. Schmid (2013);D. Andriot, A. Betz (2013)
The dual background can then by described by „dual“ metric and a bi-vector:
ds
2= (dx
1)2 + (dx
2)2 + (dx
3)2 , �
12 = Nx
3
Q-flux: Q123 = N
For the Q-flux background one obtains:
Bij(x) ! �
ij(x) =12
⇣(g �B)�1 � (g + B)�1
⌘,
g(x) ! g(x) =12
⇣(g �B)�1 + (g + B)�1
⌘�1.
T ij
T ij
Qijk = @k�ij
Donnerstag, 27. Oktober 16
12
Then one obtains from the CFT of the Q-flux background the following commutation relation among the coordinates:
⇥x
1, x
2⇤
= Np
3
In general:
⇥x
i, x
j⇤
= i
l
2s
~
I
S1k
Q
ijk (x) dx
k = i
l
3s
~ Q
ijk p
k
winding number = dual momentum
Sigma-model for non-geometric backgrounds: A. Chatzistavrakidis, L. Jonke, O. Lechtenfeld, arXiv:1505.05457
I. Bakas, D.L., arXiv:1505.04004
Donnerstag, 27. Oktober 16
13
(iv) R-flux background:
Buscher rule fails and one would get a background that is even locally not well defined.
T-duality along x
3
Donnerstag, 27. Oktober 16
13
(iv) R-flux background:
Buscher rule fails and one would get a background that is even locally not well defined.
T-duality along x
3
R-flux can be defined in double field theory:
T k
�
ij(xk) ! �
ij(xk)
T k
x
k ! xk
Rijk = 3@[k�ij]
Donnerstag, 27. Oktober 16
14
Strong constraint of DFT is violated by this background.
ds
2= (dx
1)2 + (dx
2)2 + (dx
3)2 , �
12 = Nx3
In our case we get:
R123 = NR-flux:
But it is still a consistent CFT background.
Donnerstag, 27. Oktober 16
14
Strong constraint of DFT is violated by this background.
ds
2= (dx
1)2 + (dx
2)2 + (dx
3)2 , �
12 = Nx3
Now for the R-flux background we obtain:
⇥x
i, x
j⇤
= i
l
3s
~ R
ijkpk
⇥x
i, p
j⇤
= i~�
ij,
⇥p
i, p
j⇤
= 0
=)⇥x
i, x
j, x
k⇤⌘ 1
3⇥⇥
x
1, x
2⇤, x
3⇤+ cycl. perm. = l
3sR
ijk
In our case we get:
R123 = NR-flux:
But it is still a consistent CFT background.
Donnerstag, 27. Oktober 16
14
Strong constraint of DFT is violated by this background.
ds
2= (dx
1)2 + (dx
2)2 + (dx
3)2 , �
12 = Nx3
Now for the R-flux background we obtain:
⇥x
i, x
j⇤
= i
l
3s
~ R
ijkpk
⇥x
i, p
j⇤
= i~�
ij,
⇥p
i, p
j⇤
= 0
=)⇥x
i, x
j, x
k⇤⌘ 1
3⇥⇥
x
1, x
2⇤, x
3⇤+ cycl. perm. = l
3sR
ijk
In our case we get:
R123 = NR-flux:
momentum
But it is still a consistent CFT background.
Donnerstag, 27. Oktober 16
15
● Mathematical framework to describe non- geometric string backgrounds:
Group theory cohomology. ⇒ 3-cycles, 2-cochains, - products?
Two remarks:
● The same algebra appear in the context of the magnetic monopole.
R. Jackiw (1985); M. Günaydin, B. Zumino (1985)
I. Bakas, D.Lüst, arXiv:1309.3172D. Mylonas, P. Schupp, R.Szabo, arXiv:1207.0926, arXiv:1312.162, arXiv:1402.7306.
I. Bakas, D.L., arXiv:1309.3172
Donnerstag, 27. Oktober 16
16
III) R-flux algebra from octonions
⌘ABC = 1() (ABC) = (123), (516), (624), (435), (471), (572), (673)
eAeB = ��AB + ⌘ABC eC
There exist four division algebras: over R , C , Q , O
Division algebra of real octonions : non-commutative, non-associative
O
Besides the identity, there are seven imaginary units eA
(A = 1 . . . , 7)
Donnerstag, 27. Oktober 16
17
e1 e2
e3 e4e5
e6
e7
Remark: Octonions generate a simple Malcev algebraM. Günaydin, F. Gürsey (1973); M. Günaydin, D. Minic, arXiv:1304.0410.
Fano plane mnemonic:
Donnerstag, 27. Oktober 16
18
e7
Split indices: e(i+3) = fi , for i = 1, 2, 3ei ,
and
Donnerstag, 27. Oktober 16
18
e7
Split indices: e(i+3) = fi , for i = 1, 2, 3ei ,
and
[ei, ej ] = 2✏ijkek , [e7, ei] = 2fi ,
[fi, fj ] = �2✏ijkek , [e7, fi] = �2ei ,
[ei, fj ] = 2�ije7 � 2✏ijkfk
[ei, ej , fk] = 4✏ijke7 � 8�k[ifj] ,
[ei, fj , fk] = �8�i[jek] ,
[fi, fj , fk] = �4✏ijke7 ,
[ei, ej , e7] = �4✏ijkfk ,
[ei, fj , e7] = �4✏ijkek ,
[fi, fj , e7] = 4✏ijkfk
Associator [X,Y, Z] ⌘ (XY )Z �X(Y Z)
Donnerstag, 27. Oktober 16
19
Contraction of octonionic Malcev algebra:
pi = �i�
12ei , x
i = i�
1/2
pN
2fi I = i�3/2
pN
2e7,
Donnerstag, 27. Oktober 16
19
Contraction of octonionic Malcev algebra:
pi = �i�
12ei , x
i = i�
1/2
pN
2fi I = i�3/2
pN
2e7,
�! 0
[fi, fj ] = �2✏ijkek =) [xi, x
j ] = iN✏
ijkpk
[ei, ej ] = 2✏ijkek =) [pi, pj ] = 0
[fi, ej ] = ��
ije7 + ✏
ijkfk =)
⇥x
i, pj
⇤= i�
ijI
[xi, I] = 0 = [pi, I]
[fi, fj , fk] = �4✏ijke7 =)⇥x
i, x
j, x
k⇤
= N✏
ijkI
Agrees with non-associative R-flux algebra !
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Lift of R-flux algebra to non-geometric M-theory background:
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Lift of R-flux algebra to non-geometric M-theory background:
- additional M-theory coordinatee7
⇒ Four coordinates: f1, f2, f3, e7
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Lift of R-flux algebra to non-geometric M-theory background:
- additional M-theory coordinatee7
⇒ Four coordinates: f1, f2, f3, e7
- but no additional momentum.
⇒ Three momenta: e1, e2, e3
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Lift of R-flux algebra to non-geometric M-theory background:
- additional M-theory coordinatee7
⇒ Four coordinates: f1, f2, f3, e7
- but no additional momentum.
⇒ Three momenta: e1, e2, e3
⇒ Seven dimensional phase space !
Donnerstag, 27. Oktober 16
20
What is the role of un-contracted algebra?
Lift of R-flux algebra to non-geometric M-theory background:
Will be closely related to SL(4) /SO(4) exceptional field theory.
- additional M-theory coordinatee7
⇒ Four coordinates: f1, f2, f3, e7
- but no additional momentum.
⇒ Three momenta: e1, e2, e3
⇒ Seven dimensional phase space !
Donnerstag, 27. Oktober 16
21
IV) M-theory up-lift of R-flux backgroundConsider IIA string: the duality chain splits into two
possible T-dualities:
HijkTij�! Qij
k , f ijk
Tjk�! Rijk
Donnerstag, 27. Oktober 16
21
IV) M-theory up-lift of R-flux backgroundConsider IIA string: the duality chain splits into two
possible T-dualities:
HijkTij�! Qij
k , f ijk
Tjk�! Rijk
Donnerstag, 27. Oktober 16
21
IV) M-theory up-lift of R-flux backgroundConsider IIA string: the duality chain splits into two
possible T-dualities:
HijkTij�! Qij
k , f ijk
Tjk�! Rijk
• two T-dualities ⇔ 3 U-dualities
(Need third duality along the M-theory circle to ensure right dilaton shift.)
Uplift to M-theory: add additional circle S1x
4
• 3-dim IIA flux background ⇔ 4-dim M-theory flux
background
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Use SL(5) exceptional field theory:
10 generalized coordinates:
x
A $ x
[ab] (A = 1, . . . , 10 ; a, b = 1 . . . , 5)
x
↵ = x
5↵, (↵ = 1, . . . , 4)
D. Berman, M. Perry, arXiv:1008.1763
• 4 coordinates of :T 4
• 6 dual coordinates: (x41, x
42, x
43; x21, x
31, x
32)wrapped F1 wrapped D2
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Use SL(5) exceptional field theory:
10 generalized coordinates:
x
A $ x
[ab] (A = 1, . . . , 10 ; a, b = 1 . . . , 5)
x
↵ = x
5↵, (↵ = 1, . . . , 4)
D. Berman, M. Perry, arXiv:1008.1763
• 4 coordinates of :T 4
• 6 dual coordinates: (x41, x
42, x
43; x21, x
31, x
32)wrapped F1 wrapped D2
10 of SL(5)
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Use SL(5) exceptional field theory:
10 generalized coordinates:
x
A $ x
[ab] (A = 1, . . . , 10 ; a, b = 1 . . . , 5)
x
↵ = x
5↵, (↵ = 1, . . . , 4)
D. Berman, M. Perry, arXiv:1008.1763
• 4 coordinates of :T 4
• 6 dual coordinates: (x41, x
42, x
43; x21, x
31, x
32)wrapped F1 wrapped D2
10 of SL(5) 5 of SL(5)
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Use SL(5) exceptional field theory:
10 generalized coordinates:
x
A $ x
[ab] (A = 1, . . . , 10 ; a, b = 1 . . . , 5)
x
↵ = x
5↵, (↵ = 1, . . . , 4)
D. Berman, M. Perry, arXiv:1008.1763
• 4 coordinates of :T 4
• 6 dual coordinates: (x41, x
42, x
43; x21, x
31, x
32)wrapped F1 wrapped D2
10 of SL(5)
4 of SO(4)
5 of SL(5)
Donnerstag, 27. Oktober 16
22
(i) twisted torus T 3 ⇥ S1x
4
ds
24 =
�dx
1 �Nx
3dx
2�2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , C3 = 0
(ii) R-flux background: dualise along and . x
2, x
3x
4
This leads to a locally not well defined space.
Use SL(5) exceptional field theory:
10 generalized coordinates:
x
A $ x
[ab] (A = 1, . . . , 10 ; a, b = 1 . . . , 5)
x
↵ = x
5↵, (↵ = 1, . . . , 4)
D. Berman, M. Perry, arXiv:1008.1763
• 4 coordinates of :T 4
• 6 dual coordinates: (x41, x
42, x
43; x21, x
31, x
32)wrapped F1 wrapped D2
10 of SL(5)
4 of SO(4)
6 of SO(4)
5 of SL(5)
Donnerstag, 27. Oktober 16
23
tri-vector:
SL(5) Flux-background:
g↵� =�1 + V 2
��1/3 ⇥�1 + V 2
�g↵� � V↵V�
⇤,
⌦↵�� =�1 + V 2
��1g↵⇢g��g��C⇢�� ,
ds2
7 =�1 + V 2
��1/3ds2
7 .
V ↵ =1
3!|e|✏↵���C���
@↵� = @↵� + ⌦↵��@� @
↵� =@
@x
↵�,
C. Blair, E. Malek, arXiv:1412.0635.
dual metric:
R↵,���⇢ = 4@↵[�⌦��⇢]R-flux:
Donnerstag, 27. Oktober 16
23
tri-vector:
SL(5) Flux-background:
g↵� =�1 + V 2
��1/3 ⇥�1 + V 2
�g↵� � V↵V�
⇤,
⌦↵�� =�1 + V 2
��1g↵⇢g��g��C⇢�� ,
ds2
7 =�1 + V 2
��1/3ds2
7 .
V ↵ =1
3!|e|✏↵���C���
@↵� = @↵� + ⌦↵��@� @
↵� =@
@x
↵�,
C. Blair, E. Malek, arXiv:1412.0635.
dual metric:
R↵,���⇢ = 4@↵[�⌦��⇢]R-flux:
A particular choice of R-flux breaks SL(5) to SO(4).
Donnerstag, 27. Oktober 16
24
In this way we obtain a well defined R-flux background in M-theory, which is dual to twisted torus:
R4,1234 = N
The R-flux breaks the section condition of exceptional field theory.
But it should be still a consistent M-theory background.
ds
2
7 = (dx
1)2 + (dx
2)2 + (dx
3)2 + (dx
4)2 , ⌦134 = Nx
24
Donnerstag, 27. Oktober 16
25
What are the possible conjugate momenta (or windings)?
Four coordinates:x
1, x
2, x
3, x
4
Donnerstag, 27. Oktober 16
25
What are the possible conjugate momenta (or windings)?
Four coordinates:x
1, x
2, x
3, x
4
• Consider cohomology of twisted torus:
H1(T 3 ⇥ S1, R) = R3
Donnerstag, 27. Oktober 16
25
What are the possible conjugate momenta (or windings)?
Four coordinates:x
1, x
2, x
3, x
4
Dualize to IIB: H-flux with D3-branes
• Alternatively consider Freed-Witten anomaly:
R-Flux with momentum along
R-Flux with D0 branes.
x
4p4
⇕
This is forbidden by the Freed-Witten anomaly.
⇒ No momentum modes along the direction !x
4
• Consider cohomology of twisted torus:
H1(T 3 ⇥ S1, R) = R3
Donnerstag, 27. Oktober 16
26
So we see that the phase space space of R-flux background in M-theory is seven-dimensional:
x
1, x
2, x
3, x
4 ; p1, p2, p3
Donnerstag, 27. Oktober 16
26
So we see that the phase space space of R-flux background in M-theory is seven-dimensional:
x
1, x
2, x
3, x
4 ; p1, p2, p3
4 of SO(4) 3 of SO(4)
Donnerstag, 27. Oktober 16
26
So we see that the phase space space of R-flux background in M-theory is seven-dimensional:
Missing momentum condition in covariant terms:
p↵R↵,���⇢ = 0
This condition is not the same as section condition.
x
1, x
2, x
3, x
4 ; p1, p2, p3
4 of SO(4) 3 of SO(4)
Donnerstag, 27. Oktober 16
27
V) Non-associative R-flux algebra in M-theory
Identify
Xi =12ip
Nl3/2s �1/2fi , X4 =
12ip
Nl3/2s �3/2e7 , P i = �1
2i~�ei
Donnerstag, 27. Oktober 16
27
V) Non-associative R-flux algebra in M-theory
Identify
Xi =12ip
Nl3/2s �1/2fi , X4 =
12ip
Nl3/2s �3/2e7 , P i = �1
2i~�ei
Octonionic algebra = conjectured M-theory algebra
[Pi, Pj ] = �i�~✏ijkP k ,⇥X4, Pi
⇤= i�2~Xi ,
⇥Xi, Xj
⇤= il3s
~ R4,ijk4Pk ,⇥X4, Xi
⇤= i�l3s
~ R4,1234P i ,⇥Xi, Pj
⇤= i~�i
jX4 + i�~✏i
jkXk ,⇥X↵, X� , X�
⇤= l3sR
4,↵���X� ,⇥Pi, X
j , Xk⇤
= 2�l3sR4,1234�[j
i P k] ,⇥P i, Xj , X4
⇤= �2l3sR
4,ijk4Pk ,
[Pi, Pj , Xk] = ��2~2✏ijkX4 + 2�~2�k[iXj] ,
[Pi, Pj , X4] = �3~2✏ijkXk ,
[Pi, Pj , Pk] = 0 .
Donnerstag, 27. Oktober 16
28
Remarks:
Donnerstag, 27. Oktober 16
28
Remarks:
gs : � / gs / R4 ⇒ String coupling
Natural identification of contraction parameter :�
Donnerstag, 27. Oktober 16
28
Remarks:
The non-associative M-theory algebra is not SL(5) invariant.
However the algebra is SO(4) invariant:
Lie subalgebra [Pi, Pj ] = �i�~✏ijkP k
X↵: (2, 2) of SU(2, 2)
Pi : (3, 1) of SU(2, 2)
gs : � / gs / R4 ⇒ String coupling
Natural identification of contraction parameter :�
Donnerstag, 27. Oktober 16
28
Remarks:
Further Modification compared to the string case:⇥Xi, Pj
⇤= i~�i
jX4 + i�~✏i
jkXk
The non-associative M-theory algebra is not SL(5) invariant.
However the algebra is SO(4) invariant:
Lie subalgebra [Pi, Pj ] = �i�~✏ijkP k
X↵: (2, 2) of SU(2, 2)
Pi : (3, 1) of SU(2, 2)
gs : � / gs / R4 ⇒ String coupling
Natural identification of contraction parameter :�
Donnerstag, 27. Oktober 16
V) Outlook & open questions
29
Non-associative algebras occur in M-theory at many places:
- Multiple M2-brane theories and 3-algebras
I.Bakas, E. Floratos, A. Kehagias, hep-th/9810042
J. Bagger, N. Lambert (2007)
- backgroundsSpin(7), G2
Donnerstag, 27. Oktober 16
V) Outlook & open questions
29
Non-associative algebras occur in M-theory at many places:
- Multiple M2-brane theories and 3-algebras
I.Bakas, E. Floratos, A. Kehagias, hep-th/9810042
J. Bagger, N. Lambert (2007)
What is the meaning of the odd-dimensional phase space?
Derivation from M-brane sigma modelSee also:D. Mylonas, P. Schupp, R.Szabo, arXiv:1312.1621
- backgroundsSpin(7), G2
Donnerstag, 27. Oktober 16
V) Outlook & open questions
29
Non-associative algebras occur in M-theory at many places:
Generalization to higher dimensional exceptional field theory?
- Multiple M2-brane theories and 3-algebras
I.Bakas, E. Floratos, A. Kehagias, hep-th/9810042
J. Bagger, N. Lambert (2007)
What is the meaning of the odd-dimensional phase space?
Derivation from M-brane sigma modelSee also:D. Mylonas, P. Schupp, R.Szabo, arXiv:1312.1621
- backgroundsSpin(7), G2
Donnerstag, 27. Oktober 16